An analytic approach to purely nonlocal Bellman equations arising in models of stochastic control

Given a bounded domain Ω⊂Rd and two integro-differential operators L1, L2 of the form Lju(x)=p.v.∫Ω(u(x)−u(y))kj(x,y,x−y)dy we study the fully nonlinear Bellman equation(0.1)maxj=1,2{Lju(x)+aj(x)u(x)−fj(x)}=0in Ω, with Dirichlet boundary conditions. Here, aj,fj:Ω→R are non-negative functions. We prove the existence of a non-negative function u:Ω→R which satisfies (0.1) almost everywhere. The main difficulty arises through the nonlocality of Lj and the absence of regularity near the boundary.